Type II blow-up mechanisms in a semilinear heat equation with critical Joseph–Lundgren exponent

We are concerned with blow-up mechanisms in a semilinear heat equation u_t=Δu+|u|^p−1u,x∈R^N,t>0, where p>1 is a constant. It is well known that type II blow-up does occur if N≥11 and p>p_JL, where p_JL stands for the Joseph–Lundgren exponent: p_JL={+∞,N≤10,1+[Formula presented],N≥11. On the other hand, it is also known that, if p<p_JL, type II blow-up cannot occur under mild assumptions on initial data as far as nonnegative radially symmetric solutions are concerned. It has long remained open whether or not type II blow-up occurs for the borderline case p=p_JL,N≥11. In the present article we prove constructively the existence of type II blow-up solutions for the Joseph–Lundgren critical case, thus giving an answer to this question.